- Simplipy \( \frac{6 \sqrt{2 x^{8}}-2 \sqrt{8 x^{8}}}{\sqrt{8 x^{8}}} \) without the use of a calu

Certainly! Let's simplify the expression step by step without using a calculator: \[ \frac{6 \sqrt{2 x^{8}} - 2 \sqrt{8 x^{8}}}{\sqrt{8 x^{8}}} \] ### Step 1: Simplify the Square Roots First, notice that \( x^{8} \) is a perfect square since \( x^{8} = (x^{4})^{2} \). Also, simplify the constants inside the square roots. 1. **Simplify \( \sqrt{2 x^{8}} \):** \[ \sqrt{2 x^{8}} = \sqrt{2} \cdot \sqrt{x^{8}} = \sqrt{2} \cdot x^{4} \] 2. **Simplify \( \sqrt{8 x^{8}} \):** \[ \sqrt{8 x^{8}} = \sqrt{8} \cdot \sqrt{x^{8}} = \sqrt{4 \cdot 2} \cdot x^{4} = \sqrt{4} \cdot \sqrt{2} \cdot x^{4} = 2 \sqrt{2} \cdot x^{4} \] ### Step 2: Substitute the Simplified Terms Back into the Expression Replace the square roots in the original expression with their simplified forms: \[ \frac{6 \cdot (\sqrt{2} \cdot x^{4}) - 2 \cdot (2 \sqrt{2} \cdot x^{4})}{2 \sqrt{2} \cdot x^{4}} \] ### Step 3: Distribute the Constants Multiply the constants outside the parentheses: \[ \frac{6 \sqrt{2} x^{4} - 4 \sqrt{2} x^{4}}{2 \sqrt{2} x^{4}} \] ### Step 4: Combine Like Terms in the Numerator Subtract the terms in the numerator: \[ \frac{(6 \sqrt{2} x^{4} - 4 \sqrt{2} x^{4})}{2 \sqrt{2} x^{4}} = \frac{2 \sqrt{2} x^{4}}{2 \sqrt{2} x^{4}} \] ### Step 5: Simplify the Fraction Notice that the numerator and the denominator are identical: \[ \frac{2 \sqrt{2} x^{4}}{2 \sqrt{2} x^{4}} = 1 \] ### Final Answer \[ 1 \]